Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,28,42,27,33,26,32)(23,34,43,46,37,29,36,50)(24,38,39,30,25,31,40,47)(35,48,51,49,41,45,52,44);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,35)(25,41)(26,36)(27,37)(28,43)(30,44)(31,48)(32,46)(33,34)(38,45)(39,51)(40,52)(42,50)(47,49);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,25)(22,30,33,47)(23,35,37,41)(26,39,28,40)(29,44,34,49)(31,42,38,32)(36,51,43,52)(45,46,48,50);(2,4)(7,14)(10,13)(15,20)(18,19)(21,25,27,24)(22,31,33,38)(23,35,37,41)(26,39,28,40)(29,45,34,48)(30,42,47,32)(36,52,43,51)(44,46,49,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,32,33,42)(23,36,37,43)(24,39,25,40)(29,46,34,50)(30,31,47,38)(35,51,41,52)(44,48,49,45)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089716);